We know how to find the square root of any positive real number. In a similar way we can find the square root of a negative number. The difference is that the root is not real. If the value in the radicand is negative, the root is said to be an imaginary number. The imaginary number [latex]i[/latex] is defined as the square root of negative 1.

[latex]\sqrt{-1}=i[/latex]

So, using properties of radicals,

[latex]{i}^{2}={\left(\sqrt{-1}\right)}^{2}=-1[/latex]

We can write the square root of any negative number as a multiple of [latex]i[/latex]. Consider the square root of –25.

[latex]\begin{align}\sqrt{-25}&=\sqrt{25\cdot \left(-1\right)}\\&=\sqrt{25}\cdot\sqrt{-1}\\ &=5i\end{align}[/latex]

We use [latex]5i[/latex] and not [latex]-\text{5}i[/latex] because the principal root of 25 is the positive root.

A complex number is the sum of a real number and an imaginary number. A complex number is expressed in standard form when written [latex]a+bi[/latex] where [latex]a[/latex] is the real part and [latex]bi[/latex] is the imaginary part. For example, [latex]5+2i[/latex] is a complex number. So, too, is [latex]3+4\sqrt{3}i[/latex].

Imaginary numbers are distinguished from real numbers because a squared imaginary number produces a negative real number. Recall, when a positive real number is squared, the result is a positive real number and when a negative real number is squared, again, the result is a positive real number. Complex numbers are a combination of real and imaginary numbers.

recall writing square roots in simplest form

Recall that the principal square root of [latex]a[/latex] is the nonnegative number that, when multiplied by itself, equals [latex]a[/latex]. It is written as a radical expression, with a symbol called a radical over the term called the radicand: [latex]\sqrt{a}[/latex].

To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. Use the product rule for simplifying square roots, which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite [latex]\sqrt{-75}[/latex] as [latex]\sqrt{25}\cdot \sqrt{3}\cdot \sqrt{-1} = 5  \sqrt{3}i[/latex].

 

Plot complex numbers on the complex plane

We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number we need to address the two components of the number. We use the complex plane, which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs [latex](a, b)[/latex], where [latex]a[/latex] represents the coordinate for the horizontal axis and [latex]b[/latex] represents the coordinate for the vertical axis.

Let’s consider the number [latex]-2+3i[/latex]. The real part of the complex number is [latex]–2[/latex] and the imaginary part is [latex]3i[/latex]. We plot the ordered pair [latex]\left(-2,3\right)[/latex] to represent the complex number [latex]-2+3i[/latex].

A General Note: Complex Plane

In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis.

Example: Plotting a Complex Number on the Complex Plane

Plot the complex number [latex]3 - 4i[/latex] on the complex plane.

Show Solution

The real part of the complex number is [latex]3[/latex], and the imaginary part is [latex]–4i[/latex]. We plot the ordered pair [latex]\left(3,-4\right)[/latex].

Try It

Plot the complex number [latex]-4-i[/latex] on the complex plane.

Show Solution